Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5<sup>1/3</sup> to be the cube root of 5 because we want (5<sup>1/3</sup>)<sup>3</sup> = 5<sup>(1/3)3</sup> to hold, so (5<sup>1/3</sup>)<sup>3</sup> must equal 5.

Lesson Beginning/Entry Activity

10 minutes

Begin class by handing the students a sticky note as they enter (exactly as in previous days). This time, however, you only need to use one color of sticky note because we will plot and then shift the function by slightly altering it – thus emphasizing the type of transformation that takes place in the graph. Project a graph grid on a screen, or sketch one if you do not have this supporting technology. Having done this sticky note activity several times already, the students should know exactly what to do and plot their point on the whiteboard as they are finding their seats to begin class.

As students enter, they should complete the activity introduced in the section above. This time, however, have the students plot the function f(x)=3*2^x…which is distinctly different from the previous functions. For struggling students, the order of operations may need to be revisited – especially if the class identifies multiple outliers on the graph. Showing the students their “need” before a quick review will promote student buy-in and engagement. (I hope that by this time you are able to see the value in the “identifying outliers” component of these “sticky note” introductions! Students can safely see and learn from the mistakes of their peers, and they can also predict what went wrong in the calculation of the particular output. It truly is a wonderful way to get the student’s engaged and collaborating right from the first minute of the class period! I committed myself to it each day in these lessons, and saw a dramatic DECREASE in silly mistakes each time.)

After all sticky notes have been graphed, trace the function in a marker. Then, I ask a few students to come to the board and shift the sticky notes to the function f(x)=2^x. I encourage the students to start with shifting the integer values, and estimate the location of the “nastier” exponents off of those which are easy to shift.

Once the function has been shifted, I have the students return to their thinking groups to answer the following questions:

1) How does the graph of f(x)=3*2^x differ from f(x)=2^x?

2) Describe the shape you think that f(x)=5*2^x would take. Why?

3) Describe the shape you think that f(x)=(1/2)*2^x would take. Why?

As they take 5-6 minutes to answer these questions, I typically rotate the room and take a look at the students’ homework papers (if there was an assignment) to see how they are doing. This keeps them from asking me questions, when really I want them to discuss it with their peers. I work hard to answer and student questions WITH EXTENSTION QUESTIONS… this drives them nuts, but they really learn to truly participate in the mathematical investigation.

It usually only takes the students 4-5 minutes to discuss the questions in their thinking group before I open up the class discussion. We converse and grapple with the three questions in detail before moving on in the lesson. I recommend using Geogebra during the lesson to check the students’ conjectures about the graphs.

Overview-Discovering Solutions to Exponential Equations.docx

Overview-Discovering Solutions to Exponential Equations.pdf

Lesson Middle

20 minutes

Once another layer of knowledge has been added to the students’ experience with exponential functions, then it is time to transition to solving an exponential equation for a specific value. In this lesson, I pulled up a previous PowerPoint (see Lesson #2: “Exit Slip”) and commend the students on their success for answering many of the questions correctly. Next, I asked the students HOW they decided with were larger? – Since they did so well with the activity without the ability to use a calculator.

In most cases, the class is able to come to the consensus that it is easiest to change the expressions to the same base and then compare the exponents so see which one is larger. For example, in slide #4, 4^2pi can be easily compared to 16^3 if the 16 is changed to (4^2)^3 OR 4^6. This makes it easy to see that if we have the same bases (4) – then the one with the larger exponent will always be larger (if the base b>1)! It also sets the stage for us to be able to solve exponential equations by finding like bases. As you can see, our last few lessons have put the students in a position to be successful with MP. 7!

Finally, I provide the students with a couple of concrete examples of solving an exponential equation by finding like bases. The power of the lesson actually was delivered when the students explained their thought process in a previous day’s exit slip - - discovering this provides a seamless transition to solving exponential equations.

Now, rather than asking which is larger, like we did on the previous PowerPoint Exit Slip – let’s suppose that I say: “Find the value of x that makes the statement true.”

2^x = 8^2

I have the students attempt this problem on their own, and collaborate with their thinking groups once they get their answer. I have found that through making the connection to the prior exit slip, and finding like bases, the students need little-to-no teaching at all for this concept! At this time, I usually circulate the homework assignment, which is a little more challenging. However, the students now have the tools needed to be successful! After that, it is simply a matter of encouraging them to ask questions and collaborate with their peers along the way – the true benefit of fostering a strong classroom culture of learning!

Copy of Irrational Exponents Lesson - Exit Slip.pptx

Lesson End + Homework

(I have attached sample problems from my "optional workshop" - in this lesson, most students elected to attend the extra examples. Although I kind of came up with this on the fly, I had a great deal of success calling students to the board to display the work they had done on a particular example. We learned so much from looking at the INCORRECT answers - MUCH more than even the correct ones. I enjoyed having the students locate the mistakes and learn that it is ok, and even EXPECTED to be wrong. I make a big deal about this in my classroom in order to create a safe learning environment. Without this environment, I would not have had so many hands that were willing to share!

Rather than having the students warp up the lesson, as in previous days, I take the reigns in this one and quickly summarize how utilizing the rules of exponents is one foundational way to solve an exponential equation. Prior to sending the students out the door, however, I ask them to think about/create a case when this approach would not be effective

*cough*WHEN-LIKE-BASES-CAN'T-BE-FOUND*cough*

I provide candy to the first thinking group that gets this answer. After all, this motivates our next mathematical need to know and exposese the students to the true thinking and desire of a mathematician!